No.
Angora wool comes from the Angora rabbit, of which there are several breeds. All Angora rabbits are large, some startlingly so:
Their wool grows to about 3″ (7.5cm), though some can be up to 5″ (12.5cm).
This naturally falls out every 90 days, although some owners trim their bunnies regularly:
The main reason for keeping Angora rabbits is not for their wool – it’s for competitions. After competing for a year or two, they’re retired, often becoming pets as they are very used to being handled. They’re terrific fun, a mix of dog-like rascality and cat-like snuggles.
Their wool is harvested through regular brushing, moulting, or trimming. It’s actually possible to spin directly from a bun in moult:
As long as it’s moulting, it will happily sit there and let you. If it isn’t moulting, you can expect a surprisingly painful kick in the stomach. These are not coot widdle bunnikins, they can and will draw blood.
Most owners, however, will collect wool from brushings:
Now, a number of people have mentioned PETA. It may be true that some Chinese Angora farms have poor animal protections in place, and that abuse took place. I deal with that by never buying Angora from China, or from anywhere that is even slightly suspicious – but I do that for any fibre*. However, I would like to direct you to this website, and, when you have examined it in full, I would like you to think about whether PETA is the sort of organisation from which to get your information on animal abuse.
*: this means that I only work with substandard cashmere, for example, because the truly high-quality cashmere comes from half-starved wild goats. Starvation is the reason for the ultra-fine fibre these goats produce. NB: they’re not deliberately starved. It’s because their native environment (mountain grassland) doesn’t produce enough food. Cashmere goats living in, e.g., Australia or Scotland, are well-fed, and therefore produce thicker cashmere fibre.
Quora linky.